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The rhombic dodecahedral honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is the Voronoi diagram of the face-centered cubic sphere-packing, which is the densest possible packing of equal spheres in ordinary space (see Kepler conjecture).
It consists of copies of a single cell, the rhombic dodecahedron. All faces are rhombi, with diagonals in the ratio 1:√2. Three cells meet at each edge. The honeycomb is thus cell-transitive, face-transitive and edge-transitive; but it is not vertex-transitive, as it has two kinds of vertex. The vertices with the obtuse rhombic face angles have 4 cells. The vertices with the acute rhombic face angles have 6 cells.
The rhombic dodecahedron can be twisted on one of its hexagonal cross-sections to form a trapezo-rhombic dodecahedron, which is the cell of a somewhat similar tessellation, the Voronoi diagram of hexagonal close-packing.
The trapezo-rhombic dodecahedral honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It consists of copies of a single cell, the trapezo-rhombic dodecahedron. It is similar to the higher symmetric rhombic dodecahedral honeycomb which has all 12 faces as rhombi.
It is a dual to the vertex-transitive gyrated tetrahedral-octahedral honeycomb.
The rhombic pyramidal honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. John Horton Conway calls it a truncated tetraoctahedrille.
It is dual to the cantic cubic honeycomb:
M. C. Escher, MathWorld, Mosaic, Spain, Ancient Rome
Area, Euclid, Parallelogram, MathWorld, Rectangle
Tessellation, Cubic honeycomb, Geometry, Euclidean geometry, Non-Euclidean geometry
Johnson solid, Rhombus, Trapezoid, Vertex configuration, List of spherical symmetry groups
Cuboctahedron, Tesseract, Cube, Face configuration, Rhombic triacontahedron
Tessellation, Tetrahedron, Vertex figure, Convex uniform honeycomb, Octahedron
24-cell, Tesseract, Schläfli symbol, Vertex figure, Tessellation
Rhombus, Rhombic antenna, Rhombic dodecahedron, Rhombic triacontahedron, Rhombic dodecahedral honeycomb